Graph-theoretic studies of combinatorial optimization problems

Abstract

Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. It is the study of pair-wise relationships between objects. Each object is represented using a vertex, and in case of a relationship between a pair of vertices, they will be connected using an edge.In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment problem using its underlying hypergraphs. It will be shown how the MAP can be represented by a k-partite graph and how any solution to MAP is associated to a k-clique in the respective k-partite graph. Two heuristics are proposed to solve the MAP and computational studies are performed to compare the performance of the proposed methods with exact solutions. On the heels of chapter 2, a new branch-and-bound method is proposed to solve the problem of finding all k-cliques in a k-partite graph in chapter 3. The new method utilizes bitsets as the datastructure to represent graph data. A new pruning method is introduced in BitCLQ, and CPU instructions are used to improve the performance of the branch-and-bound method. BitCLQ gains up to 300% speed up over existing methods. In chapter 4, two new heuristics to solve decomposable cost MAP's are proposed. The proposed heuristic are based on the partitioning of the underlying graph representing the MAP. In the first heuristic method, MAP's are partitioned into several smaller MAP's with the same dimensial-2 ity and smaller cardinality. The second heuristic works in the same fashion. But instead of partitioning the graph along the elements, graphs are divided into smaller graphs with the same cardinality but smaller dimensionality. The heuristics are then used in exact branch and bound methods and numerical comparison of the resulting method is provided. Maximum Clique problem entails finding the size of the largest clique contained in a graph. General branch-and-bound methods to solve MCQ use graph coloring to find an upper bound on the size of the maximum clique. In chapter 5, a branch and bound algorithm is proposed for the maximum clique problem. that is based on the method of 5. Chapter 6 contains an application of a graph theory in solving a risk management problem. The mixed-integer mathematical model to formulate a risk-based network is provided. It will be shown that an optimal solution of the model is a maximal clique in the underlying graph representing the network. Abstract Approved: Thesis SupervisorABSTRACT Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. It is the study of pair-wise relationships between objects. Each object is represented using a vertex, and in case of a relationship between a pair of vertices, they will be connected using an edge.In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment p...